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Structural identification of gravity-type caisson structure via vibration feature

analysis

Article in SMART STRUCTURES AND SYSTEMS · February 2015

DOI: 10.12989/sss.2015.15.2.259

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Smart Structures and Systems, Vol. 15, No. 2 (2015) 259-281

DOI: http://dx.doi.org/10.12989/sss.2015.15.2.259 259

Copyright © 2015 Techno-Press, Ltd.

http://www.techno-press.org/?journal=sss&subpage=8 ISSN: 1738-1584 (Print), 1738-1991 (Online)

Structural identification of gravity-type caisson structure via vibration feature analysis

So-Young Leea, Thanh-Canh Huynhb and Jeong-Tae Kim

Department of Ocean Engineering, Pukyong National University, Busan 608-737, Korea

(Received October 9, 2014, Revised January 13, 2015, Accepted January 16, 2015)

Abstract. In this study, a structural identification method is proposed to assess the integrity of gravity-type caisson structures by analyzing vibration features. To achieve the objective, the following approaches are implemented. Firstly, a simplified structural model with a few degrees-of-freedom (DOFs) is formulated to represent the gravity-type caisson structure that corresponds to the sensors’ DOFs. Secondly, a structural identification algorithm based on the use of vibration characteristics of the limited DOFs is formulated to fine-tune stiffness and damping parameters of the structural model. Finally, experimental evaluation is performed on a lab-scaled gravity-type caisson structure in a 2-D wave flume. For three structural states including an undamaged reference, a water-level change case, and a foundation-damage case, their corresponding structural integrities are assessed by identifying structural parameters of the three states by fine-tuning frequency response functions, natural frequencies and damping factors.

Keywords: structural identification; gravity-type caisson; simplified vibration model; wave flume; stiffness; damping, modal parameters

1. Introduction

Over the last two decades, many gravity-type caisson structures have been constructed in Korea

for the use of breakwater and quay wall. Caisson structures stand on foundation mound

constructed above seabed and stabilized by their own weights. Most body of harbor caisson is

submerged in sea water to keep the tranquility in harbor by dissipating wave energy from open sea.

The caisson structures cannot be ensured for its safety upon repeatedly experiencing wave forces

that are stronger than design wave forces. Due to the severe loading conditions, those structures

may result in systematic changes deviated from the as-built designs.

Recent years, the integrity of harbor caisson structures becomes more important issue due to

severe environmental phenomena and extreme events like large-scale typhoon or ship collision.

For instance, typhoon Maemi hit the southern part of the Korean Peninsula in 2003 and it resulted

in failure of gravity-type caisson structures. The gravity-type caisson structure is inevitably

damaged due to local failures or global instability problems which are mostly attributed to

Corresponding author, Professor, E-mail: [email protected] a Research Assistant

b Graduate Student

mailto:[email protected]

So-Young Lee, Thanh-Canh Huynh and Jeong-Tae Kim

problems in foundation-structure interface. For instance, upon experiencing extreme wave forces,

local defects such as cavity and displacement occurred in the foundation mound are eventually

transformed into structural instable states by weakening global functionality against settlement,

overturning and sliding of the structure (Oumeraci 1994, Goda 1994, Takahashi et al. 2000, Lee et

al. 2009). Therefore, there is a need to identify the structural performance of the damaged caisson

system which is locally defected in the foundation-structure interface.

Many researchers have worked on the so-called vibration-based structural health monitoring

techniques, which implement vibration characteristics for structural integrity assessment, in the

field of civil engineering (Adams et al. 1978, Stubbs and Osegueda 1990, Li et al. 2004, Jang et al.

2012, Huang and Nargarajaiah 2014, You et al. 2014). Also, many researchers have worked on

developing structural identification methods such as modal sensitivity method, modal flexibility

method, genetic algorithm, neural network, etc. (Aktan et al. 1997, Kim et al. 2003, Yun et al.

2009, Li et al. 2014, Gul et al. 2014). Despite those research efforts, coastal and harbor caisson

structures have been treated by only a few research attempts, which include vibration response

analyses and structural integrity estimation (Yang et al. 2001, Kim et al. 2005, Lee et al. 2013, Yi

et al. 2013).

For the vibration-based structural health monitoring of gravity-type caisson structures, at least a

few research needs exist. There are needs to experimentally analyze their vibration characteristics

via the limited accessibility and to estimate their structural integrities by using changes in the

vibration characteristics (Lee et al. 2013, Le et al. 2013). Related to these research needs, this

study has been motivated to resolve three important issues: the first one is to construct a simplified

structural model which can represent the caisson structural system with limited

degrees-of-freedom (DOFs) adopted for vibration measurement; the second one is to estimate

structural integrity of the caisson system via the simplified structural model; and the third one is to

examine effects of water-level change and foundation damage on the structural integrity of the

caisson system.

In this study, a structural identification method is proposed to assess the integrity of

gravity-type caisson structures by analyzing vibration features. To achieve the objective, the

following approaches are implemented. Firstly, a simplified structural model with a few DOFs is

formulated to represent the gravity-type caisson structure that corresponds to the sensors’ DOFs.

Secondly, a structural identification algorithm based on the use of vibration characteristics of the

limited DOFs is formulated to fine-tune the structural model. Structural stiffness and damping are

selected as two key parameters to be identified for the model update. Finally, experimental

evaluation is performed on a lab-scaled gravity-type caisson system in a 2-D wave flume. Three

test scenarios are selected as an undamaged reference, a water-level change state, and a

foundation-damage state. For each case, the corresponding structural integrity is assessed by

identifying its structural parameters by fine-tuning frequency response functions, natural

frequencies and damping factors.

2. Simplified structural model of gravity-type caisson system

Since as early as 1990, many researchers have worked on structural modeling of caisson

structural systems (Smirnov and Moroz 1983, Marinski and Oumeraci 1992, Goda 1994,

Oumeraci and Kortenhaus 1994, Vink 1997, Zhang 2006, Wang et al. 2006). However, the

proposed models would not fit into the usage for structural identification because most of them

260


require certain DOFs to be guaranteed for the function of the models. In another words, the

limitation in measurable DOFs of in-field caisson systems, which is mainly due to the lack of

accessibility, hinders the model’s usage for structural integrity assessment. Moreover, the modeling

becomes complicated as it deals with interlocking conditions since it requires additional DOFs of

vibration responses.

In order to overcome the above-mentioned limitations, in this study, a simplified structural

model is formulated to represent the gravity-type caisson structure with a few DOFs that

correspond to the sensors’ DOFs. It is noted that this study is limited to a planar model of a system

of three interlocked caissons on the basis of the existing models proposed by Goda (1994),

Oumeraci and Kortenhaus (1994), and Vink (1997). As shown in Fig. 1(a), the caisson system is

subjected to an impulsive breaking wave force that results in forced vibration responses.

Assuming that the wave action is perpendicular to the caisson array axis (i.e., y-direction),

vibration responses along with the y-direction are dominant (Lee et al. 2011, 2012, Yoon et al.

2012). This study is limited to take into account only for the horizontal displacement (e.g., sway

motion) which is measurable in the caisson system by the use of uni-directional sensor orientation.

As shown in Fig. 1(b), caissons are treated as rigid bodies on elastic half-space foundations which

can be described via the horizontal springs and dashpots (Huynh et al. 2013). Horizontal spring

and dashpots with respect to three caissons are defined, respectively. To represent the condition of

the interlocking mechanism, springs and dashpots are also simulated between adjacent caisson

units.

By equating to the equilibrium conditions of the free-body diagrams, as shown in Fig. 1(b), the

horizontal displacement can be formulated in matrix form as

[M]{�̈�} + [C]{�̇�} + [K]{𝑢} = {F} (1)

in which [M], [K] and [C] represent the mass matrix, stiffness matrix, and damping matrix,

respectively; and F is the vector of external force. As shown in Fig. 1(b), structural properties of the caisson system are defined as follows: mj is the total horizontal mass of the j

th caisson; kFj

and cFj, respectively, represent the horizontal spring and damping coefficients of the jth caisson’s

foundation (j=1-3); kSk and cSk, respectively, represent the horizontal spring and dashpot of the nth

shear-key connection (n=1-4); the symbols 𝑢�̈�, 𝑢�̇� and 𝑢𝑗 are, respectively, the horizontal

acceleration, velocity and displacement of the jth caisson; and Pj(t) is the external force placed at

the center of gravity of the jth caisson.

When the caisson is oscillated by an impulse load, its boundary media (i.e., soil and water) are

also forced to move with the structure. Therefore, the total horizontal mass of the jth caisson (𝑚𝑗)

includes not only the mass of the caisson (𝑚𝑗𝑐𝑎𝑖) but also the contribution from horizontal

hydrodynamic mass (𝑚𝑗ℎ𝑦𝑑

) and horizontal geodynamic mass (𝑚𝑗𝑔𝑒𝑜

), as follows

cai hyd geo

j j j jm m m m (2)

To calculate the horizontal hydrodynamic mass (hydjm ), the equation presented by Oumeraci

and Kortenhaus (1994) is used

𝑚𝑗ℎ𝑦𝑑

= 0.543𝜌𝑤𝐿𝑗𝐻𝑤𝑗2 (3)

261


in which the quantities Lj and 𝐻𝑤𝑗 are the jth caisson’s length and the water level, respectively, as

shown in Fig. 1(a); and the quantity w is the mass density of sea water. According to Richart et al.

(1970), the horizontal geodynamic mass can be computed as

3/ 2

0.76 /(2 )j jgeo

j s

B Lm

(4)

where s and are, respectively, the mass density and Poisson’s ratio of the foundation soil; and Bj is the j

th caisson’s width, as sketched in Fig. 1(a).

As adopted by Goda (1994) and Vink (1997), the horizontal spring constant (kFj) of the elastic

foundation is modeled as the function of the horizontal modulus of subgrade reaction (b), as

follows

𝑘𝐹𝑗 = 𝑏𝐿𝑗𝐵𝑗 (5)

in which the modulus of subgrade reaction (b) is available for various soil types as reported by

Bowles (1996). Note that the modulus has the unit of pressure per length.

Next, the stiffness of the middle shear keys is modeled as kS2 = kS3 by assuming that caisson

segments are designed with the uniform linking capacity. However, it is assumed that the stiffness

of the last shear keys (i.e., kS1 and kS4) is smaller than that of the middle shear keys (i.e., kS2 and

kS3).

𝑘𝑆1 = 𝑘𝑆4 = 𝑝𝑘𝐹1 and 𝑘𝑆2 = 𝑘𝑆3 = 𝑞𝑘𝐹2 (6)

in which p and q are stiffness ratios with respect to 𝑘𝐹. Note that the theoretical basis for the shear-key’s stiffness is relatively small as compared to the foundation’s stiffness, since it depends

on the linking capacity between contacted units in the real caisson breakwater (Oumeraci et al.

2001).

The Rayleigh damping, which is often used in the dynamic mathematical model, is used to

simulate the energy dissipation in the caisson system. The Rayleigh damping is assumed to be

proportional to the mass and stiffness matrices (Wilson 2004)

KMC (7)

in which is the mass-proportional damping coefficient; and is the stiffness-proportional damping coefficient. Due to the orthogonality condition of the mass and stiffness matrices, this

equation can be rewritten as

𝜉𝑖 =1

2𝜔𝑖𝛼 +

𝜔𝑖

2 (8)

in which i is the critical-damping ratio for ith mode; and i is the i

th natural frequency. If the

damping ratios corresponding to two specific frequencies (e.g., e and f) are known, the two

Rayleigh damping factors (i.e., and ) can be evaluated from the following equation

[𝜉𝑒𝜉𝑓

] =1

2[

1

𝜔𝑒𝜔𝑒

1

𝜔𝑓𝜔𝑓

] [𝛼𝛽] (9)

262


(a) Caisson system of three units

(b) Simplified model of caisson system

Fig. 1 A caisson system of three units and its simplified model (Huynh et al. 2013)

When damping ratios for both frequencies are set to an equal value, e = f = , the Rayleigh damping factors are calculated as (Wilson 2004)

𝛽 =2𝜉

𝜔𝑒+𝜔𝑓 and 𝛼 = 𝜔𝑒𝜔𝑓𝛽 (10)

3. Vibration-based structural identification algorithm

For the performance assessment of the gravity-type caisson system, our interest is to analyze

exactly how the structure will respond under given excitation conditions and, especially, with what

amplitudes. This will depend on the structure’s inherent properties (such as stiffness, mass and

damping) and also on the nature and magnitude of the applied excitation (Ewins 2000). For most

of real harbor caisson structures, however, it is almost impossible to impose the excitation enough

for meaningful responses. An alternative way is to choose a standard excitation and to describe the

responses as a set of frequency response functions (FRFs) based on a unit-amplitude sinusoidal

force applied to the structure. There exists the interdependence between the structural parameters

and the modal responses (e.g., FRFs, natural frequency and damping); therefore, the structural

parameters (e.g., stiffness, mass and damping) can be identified from the inverse analysis of the

263


measured response properties. Once accurately identified, the structural model may represent the

estimation of field responses corresponding to any design forces on the real structure. Based on

this concept, in this study, a structural identification algorithm is designed for the gravity-type

caisson structure system.

3.1 Vibration response analysis

As stated in Eq. (1), a structural model is represented by structural dynamic characteristics such

as stiffness, mass, and damping properties. Its acceleration responses depend on the structural

characteristics and it can be defined as

1

M C Kt t tu F u u

(11)

where tu , tu and tu represent the displacement, velocity, and acceleration vectors, respectively;

[M], [K] and [C] represent the mass matrix, stiffness matrix, and damping matrix, respectively;

and F is the vector of external force. The acceleration response provides an understanding of the dynamic characteristics that represent the structural behaviors. In this study, piezoelectric

accelerometers are utilized to measure the acceleration responses. The piezoelectric material in the

sensor acts as a spring and connects the base of the accelerometer to a seismic mass. When an

input is introduced to the base, a force is created on the piezoelectric element that is proportional

to the applied acceleration and the size of the seismic mass.

For the forced response of a damped structural system, the general receptance FRF can be

simplified in a complex form as follows (Ewins 2000)

1

2( ) K C Mi

(12)

in which the receptance ( ) is defined as force to displacement response ratio in frequency

domain, so that Eq. (11) can be interpreted as Eq. (12). Due to the orthogonal condition of mass

and stiffness matrices, the damped system can be evaluated for the ith vibration mode, as follows

2 21i i i (13a)

𝜉𝑖 = 𝛽𝜔𝑖/2 + 𝛼𝜔𝑖/2 (14b)

where i is the ith critical-damping ratio; i is the i

th undamped natural frequency; and i is

the ith damped natural frequency. If the damping ratios (e.g., e and f) corresponding to two

specific undamped natural frequencies (e.g., e and f) are known, the two Rayleigh damping

factors (i.e., and ) can be evaluated from Eq. (10). As shown in Fig. 2(a), the simplest peak-picking method is adopted for analyzing measured

FRF data to obtain the simplified model of the target caisson structure described in Fig. 1. The

resonance peak is detected on the FRF plot and the frequency of the maximum response is taken as

the ith natural frequency (i). Also, the i

th modal amplitude can be taken at the maximum response.

Next, the local maximum value of measured FRF is noted as H . The frequency bandwidth of

the function for a response level of 2H is determined as the two half-power points, a and

264


b . Then the damping of the ith mode can be experimentally estimated from the following formula

(Ewins 2000)

2 2 22i a b i (15)

Assume there are two different systems, i.e., experimental state (noted as ‘E’) and analytical

model (noted as ‘A’), as shown in Fig. 2(b). Then their systematic differences in structural

parameters can be quantified by estimating changes in vibration features like natural frequencies

and FRFs. In this study, the change in eigenvalues is used to represent the change in stiffness

parameter. Also, the change in FRFs is used for the change in damping parameter.

3.2 Structural identification algorithm

Kim and Stubbs (1995) proposed a system identification method to identify a realistic

theoretical model of a structure by using vibration modal parameters. Based on their method, in

this study, structural identification is mainly performed in two steps: first, an initial structural

model is designed based on geometrical and material properties and boundary conditions; and next,

structural parameters (e.g., stiffness and damping) are identified by fine-tuning modal parameters.

Suppose *jp is an unknown structural parameter (i.e., mass, stiffness, or damping parameters)

of the thj member of a structure for which modal parameters (i.e., natural frequencies and

damping coefficients) are known for M vibration modes. Also, suppose jp is a corresponding

known structural parameter of the thj member of an analytical model for which the

corresponding set of modal parameters are known for the same M vibration modes.

Relative to the target structure, the structural parameter of the thj member of the analytical

model is defined as follows

*

j j jp p p (16)

in which jp is the variation of the structural parameter of the thj member which results in

changes of modal parameters. The sensitivity ijS of the thi damped eigenvalue 2i with

respect to the thj structural parameter jp is defined as follows

2

2

jiij

j i

pS

p

(17)

in which the damped eigenvalue is defined as 2 2 2(1 )i i i . Since the sensitivity ijS is

related to the combination of the undamped natural frequency and damping coefficient, a

simplified approach is needed to analyze the sensitivity with respect to each parameter.

Stiffness Parameter Identification

By fixing the damping coefficient as constant from Eq. (17), the stiffness sensitivity SijS of the thi undamped eigenvalue 2i with respect to the

thj stiffness parameter jk is defined as

2

2

jS iij

j i

kS

k

(18)

265


(a) FRF plot (b) Experiment versus analytical model

Fig. 2 FRF analysis by peak picking method

where jk is the first order perturbation of jk which produces the variation in eigenvalue

2

i .

The fractional change in stiffness parameters of NE members may be obtained using the following equation (Kim and Stubbs 1995)

1

S S

j jk k S Z

(19)

where j jk k is a 1NE matrix containing the fractional changes in stiffness parameters

between the analytical model and the target structure; SZ is a 1M matrix containing the

fractional changes in eigenvalues between two systems; and SS is a NEM dimensionless

stiffness sensitivity matrix. The convergence between the measured and analytical eigenvalues is

estimated by the fractional changes in eigenvalues between the experimental state and the

analytical model, as follows

2 2

, ,

2

,

i m i aS

i

i a

Z

(20)

in which 2,i m and

2

,i a are the thi eigenvalues of the measured target structure and the

analytical model, respectively. The completeness of the model update is identified as the

convergence approaches to a tolerance value.

Damping Parameter Identification

By fixing the undamped eigenvalue as constant from Eq. (17), the damping sensitivity DijS of

the thi modal damping coefficient 2i with respect to the

thj structural damping parameter

jc is defined as

)(

H

2H

ia b

)(

EH

Ei )(a b

AH

Ai )(

2H

266


2

2 1

jD iij

j i

cS

c

(21)

where jc is the first variation of jc which produces the variation in damping coefficient i .

The fractional change in structural damping parameters of NE members may be obtained as follows

1

D D

j jc c S Z

(22)

where j jc c is a 1NE matrix containing the fractional changes in damping parameters

between the analytical model and the target structure; DZ is a 1M matrix containing the fractional changes in damping coefficients between two systems; and

DS is a NEM

dimensionless damping sensitivity matrix. The fractional change in damping coefficients between

the experimental state and the analytical model is estimated as follows

2 2

, ,

2

, 1

i m i aD

i

i a

Z

(23)

in which 2,i m and

2

,i a represent the thi damping eigenvalues of the measured target structure

and the analytical model, respectively.

The reliability of Eq. (23) depends rather upon the quality of experimental damping values.

Moreover, the accuracy of damping coefficients extracted from real structures is not guaranteed for

most of civil engineering structures. So there should be an alternative way to identify the

convergence of damping parameters for the gravity-type caisson system, in which experimental

damping estimation would not be accurate. Our choice is power spectral density (PSD) which

efficiently represents all of modal characteristics and also minimizes the noise effects. The PSD is

utilized to examine the change in damping property as much as the change in modal amplitude.

The PSD ( )S f can be calculated by Welch’s procedure

2

1

1( ) ( , )

dn

i

id

S f Y f Tn T

(24)

in which ( , )iY f T is FFT result of the acceleration signal. To estimate the change in the

frequency response due to the change in structural properties, the root mean square deviation

(RMSD) of PSDs is calculated as follows

2

2

( ) ( )( , )

( )

E A

A E

A

S f S fRMSD S S

S f

(25)

where ( )AS f and ( )ES f are the PSDs of the analytical model and the experimental state,

267


respectively. The convergence between the measured and analytical damping properties is

identified as the RMSD close to zero.

4. Experimental vibration analysis of lab-scale caisson structure

4.1 Description of test Set-up

As shown in Fig. 3, a lab-scale caisson model is selected as the target structure. Four main

subsystems include caisson body, cap concrete, two adjacent blocks, and foundation mound. The

caisson’s height and width are square as 34 cm and 34 cm; it also includes the empty inner space

with vertical wall of 5 cm thickness. The cap concrete’s height is 10 cm at the maximum. Bottom

dimension of the cap concrete is the same as that of the caisson. Due to the limited width of the

2-D wave flume, two adjacent caissons were replaced by adopting concrete blocks which have 6

cm thickness to make the main caisson body fitted into the side wall of the wave flume. Each

caisson body has shear keys for interlocking between adjacent caisson bodies.

Vibration responses were measured by a data acquisition system consisting of accelerometers

(PCB 393B04 model with ± 5g of measureable range and 1 V/g of sensitivity), a signal conditioner, a DAQ card, and a laptop. As shown in Fig. 3, the accelerometers were installed at the

top of the cap concrete to measure the vibration response of the caisson. For the design of sensor

locations in the lab caisson, the limited accessibility of coastal and harbor caisson systems was

considered since the partially-submerged on-site condition imposed restrictions on the possibility

of extracting modal parameters from the entire caisson geometry.

In this study, rigid body motions were selected as the target behaviors of the test caisson. Note

that the rigid body motion is defined when deformation of caisson is much smaller than

deformation of foundation mound. It has been reported that the rigid body motion is sensitive to

the variation of foundation integrity (Lee et al. 2012, Huynh et al. 2013). According to the

previous studies by Ming et al. (1988) and Lee et al. (2012), the first a few modes corresponding

to the rigid body motions could be extracted from the field tests and even from lab tests for the

caisson structure.

Fig. 3 Test setup in the 2-D wave flume

Cap concrete

Caisson

TTP

Foundation

mound Adjacent block

y

z

x

268


To extract the vibration characteristics corresponding to the rigid body motions of the test

caisson, sensors were installed at four points along with y-and z-axes. Acceleration signals were

acquired for 20 seconds at 500 Hz of sampling rates. Impact excitations that simulate ship

collisions were implemented by using a rubber hammer applied on the front wall of the caisson.

The impact was manually applied by a test implementer. Further details on the lab tests are

described in Lee et al. (2013).

4.2 Test scenarios and vibration responses Mainly two different conditions were considered for estimating their effects on structural

properties and vibration characteristics of the lab-scale caisson system. The first condition is

water-level variation from high water level (HWL) to low water level (LWL) and the second one is

foundation damage like the cavity in found mound. As the test scenarios, three cases were select as:

(1) an undamaged reference structure with HWL, (2) an undamaged structure with water-level

change into LWL, and 3) a foundation-damaged structure with HWL (i.e., no water-level change).

To simulate the water-level variation, two water-levels were designed in 2-D wave flume. As

indicated in Fig. 3, the HWL and the LWL were selected based on on-site tidal conditions near

Busan, Korea. They vary about 1.2 m during high and low tides and it was scaled down as 44mm

as the depth variation. The caisson was kept undamaged during the tests.

Next, damage was inflicted into the foundation mound of the caisson system and changes in

modal parameters were estimated. The water level was fixed as the HWL during the test. The

occurrence of the foundation damage is typically caused by dislocation of wave-dissipating blocks

and subsequent scouring induced by repeated extreme wave actions. In this study, the damage was

simulated by removing some parts of armor blocks and TTP blocks from the foundation mound,

which caused 18.7% loss out of the bottom area of the caisson.

The test caisson was excited by hammer impact and acceleration responses were measured by

the vibration measurement system. Fig. 5 shows the y-directional and z-directional vibration

responses measured at the acquisition point 1. Note that the y-direction corresponds to wave

incident direction. The maximum y-directional acceleration was about 0.25 g with short impulse

duration; meanwhile, the maximum z-directional acceleration was less than 0.1 g.

The peak picking method was implemented to extract frequency responses and modal

parameters for the three test cases. For the case of ‘the undamaged reference structure with HWL’,

the power spectral density was extracted from the acceleration signal measured from the sensor

1-y (see Fig. 4). For the analysis of the reference caisson with HWL, the first two modes were

extracted by using the stochastic subspace identification (SSI) method and the frequency-domain

decomposition (FDD) method (Yi and Yun 2004, Lee et al. 2013). The extracted natural

frequencies are sketched as shown in Fig. 6. The first two natural frequencies were observed at

about 17 Hz and 42 Hz in both results of the FDD method and the SSI method. The first two

damping coefficients were about 6.9% and 2.7%, respectively.

For all three test cases, natural frequencies and damping coefficients were extracted as listed in

Table 1. From the results, it is observed that the natural frequencies increased as the water level

decreased (i.e., as HWL turns to LWL) but they decreased as the foundation damage occurred. It is

also observed that damping ratios were inconsistent in mode 1 and mode 2. In mode 1, damping

ratios were decreased by water-level decrease but increased by damage occurrence. In mode 2,

however, there was almost no change. Also, the change in damping ratio due to the ground damage

seemed to be overestimated.

269


Fig. 4 Orientation of sensors and excitation point (Lee et al. 2013)

(a) Y-direction (b) Z-direction

Fig. 5 Acceleration responses at acquisition point 1 excited by hammer impact

Table 1 Experimental modal parameters of test caisson system for three test cases

Test scenario Natural frequency (Hz)

Damping ratio (%)

Peak-Picking SSI

Mode 1 Mode 2 Mode 1 Mode 2 Mode 1 Mode 2

Undamaged

HWL 17.33 41.99 6.89 2.74 7.14 4.00

Undamaged

LWL 19.78 44.68 5.26 2.64 6.20 N/A

Damaged

HWL 15.39 42.24 12.08 2.89 6.89 3.50

1-y

1-z

2-y

2-z3-y

3-z

4-y

4-z

y

z

x

: Acquisition point

: Y-directional impact point

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5

Time (s)

Acc

eler

atio

n (

g)

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5

Time (s)

Acc

eler

atio

n (

g)

270


To check the reliability of the extracted damping ratios from the peak-picking method, the SSI

method was also utilized to extract damping ratios. As listed in Table 1, damping ratios extracted

by the SSI method were also inconsistent as compared to those by the peak-picking method. Due

to the inconsistent quality of experimental damping values, the alternative way utilizing PSDs and

RMSD (as described in Eqs. (24) and (25)) should be utilized to identify the convergence of

damping parameters of the caisson structure system.

5. Vibration-based structural identification of lab-scale caisson structure

5.1 Structural identification for reference structure

Initial Structural Model

The structural model of ‘the undamaged reference structure with HWL’ was initially established

by using geometry of the test structure and the experimental modal parameters. The total

horizontal masses were computed with 0.34 m of the water height as m1 = m3 = 22.95 kg and m2 =

138.06 kg by using Eqs. (2)-(4). The diagonal components of the M matrix consisted of the mass

of concrete, the mass of water filled, the added mass of sea water, and the added mass of

foundation soil.

The modulus of sub-grade reaction of the foundation mound, b, was selected as 9.6 MN/m for

medium dense sand (Bowles 1996). By using Eq. (5), the spring constants of the foundation

mound were calculated as 𝑘𝐹1 = 𝑘𝐹3 =1.824x105

N/m and 𝑘𝐹2 =1.110x106

N/m. By assuming

stiffness ratios as p = 1, and q = 0.5 for the initial model, the stiffness of the middle and last

shear-keys were obtained as 𝑘𝑆1 = 𝑘𝑆4 = 1.824x105

N/m and 𝑘𝑆2 = 𝑘𝑆3 = 5.549x105

N/m,

respectively. As the initial damping parameter, the two Rayleigh damping coefficients were

computed from Eq. (13) by using the first two natural frequencies (ω1 = 17.33 Hz, ω2 = 41.99 Hz)

and the damping ratio (ξ1 = 6.89 %, ξ2 = 2.74 %). In Table 2, the calculated K and C matrices of the

initial structural model are summarized in forms of relative values.

To solve the equation of motion (see Eq. (1)), the Runge-Kutta scheme was utilized as

supported in Matlab 7.9.0.529. Time interval was set as 0.001 second for the calculation of

vibration responses. As shown in Fig. 7, the acceleration and PSD responses of the initial structural

model were compared with those of the experimental state of ‘the undamaged caisson with HWL’.

It is observed that the two acceleration responses show differences in dissipation and oscillating

period. It is also observed that the two PSD curves are quite different in the peaks and the

dissipation slopes. Note that the initial impact force was assumed as 1N in the numerical

simulation. It was assumed that relationship between acceleration response and impact force is

linear. Then, impact force of the simulation was determined to the value which gives similar

response to experimental response in the maximum acceleration.

Model Update: Stiffness Parameter

The stiffness parameters were updated by using the trial-and-error method to improve the

convergence of the structural model to the experimental state. The stiffness ratios, p and q, of Eq.

(6) controlled to adjust the peak frequencies of power spectral density. As the result of the

stiffness parameter update, the stiffness ratios were selected as p=1.5 and q=0.8, so that the

stiffness of the shear-keys was updated as 𝑘𝑆1 = 𝑘𝑆4 =2.736x105 N/m and 𝑘𝑆2 = 𝑘𝑆3 =8.880x10

5

N/m, respectively. The calculated K matrix and C matrix with the updated stiffness parameters are

271


summarized in Table 2.

As shown in Fig. 8, the acceleration and PSD responses of the stiffness-updated model were

compared with those of the experimental state. It is observed that the two acceleration responses

show very close oscillating period but difference in dissipation. It is also observed that the two

PSD curves are similar in the natural frequencies but different in the dissipation slopes. Note that

the stiffness parameters were updated using the trial-and-error method. In general, the manual

tuning might not be objective; however, the process might be one of the choices due to the

complexity of the structural and response parameters.

Model Update: Damping Parameter

The damping parameters were updated by using the trial-and-error method to improve the

convergence of the structural model’s dissipation of vibration responses to that of the experimental

state. The damping parameters, ξ1 and ξ2, were used as updating parameters to adjust the peak

dissipation of power spectral density. As the result of the damping parameter update, the damping

ratios are selected as ξ1=15.17%, ξ2=10.98%, respectively. The calculated K and C matrices with

the updated damping parameters are summarized in Table 2.

Fig. 6 SSI method’s stabilization charts and FDD method’s singular values for the undamaged reference

caisson with HWL

Table 2 Estimated structural parameters for ‘the undamaged reference caisson with HWL’

Stiffness ratio K matrix C matrix

Initial 𝑘𝑆1 = 𝑘𝑆4 = 𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.5𝑘𝐹2 [

0.92 −0.55 0−0.55 2.22 −0.55

0 −0.55 0.92] × 106 [

0.34 0.01 00.01 2.07 0.01

0 0.01 0.34] × 103

Stiffness

update

𝑘𝑆1 = 𝑘𝑆4 = 1.5𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.8𝑘𝐹2 [

1.34 −0.89 0−0.89 2.89 −0.89

0 −0.89 1.34] × 106 [

0.33 0.01 00.01 2.06 0.01

0 0.01 0.33] × 103

Damping

update

𝑘𝑆1 = 𝑘𝑆4 = 1.5𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.8𝑘𝐹2 [

1.34 −0.89 0−0.89 2.89 −0.89

0 −0.89 1.34] × 106 [

1.22 −0.38 0−0.38 5.10 −0.38

0 −0.38 1.22] × 103

-15

-10

-5

0

5

10

15

0

50

100

150

200

250

0 10 20 30 40 50 60

Frequency (Hz)

Ob

serv

atio

nar

yo

rder

Sin

gu

lar

val

ues

in

FD

D

Mode 1 Mode 2

272


(a) Acceleration response (b) Power spectral density

Fig. 7 Vibration responses of initial model for ‘the undamaged caisson with HWL’


Fig. 8 Vibration responses of stiffness-updated model for ‘the undamaged caisson with HWL’

As shown in Fig. 9, the acceleration and PSD responses of the damping-updated model were

compared with those of the experimental state. It is observed that the two acceleration signals are

well-matched. Two responses show very close oscillating period and also in dissipation. It is also

observed that the two PSD curves are well-matched both the peak frequencies and the dissipation

slopes. Note that the primary motivation of updating the damping parameter is to identify the

energy dissipation capacity, so that the update damping parameters represents the energy

dissipation capacity.

The accuracy of the structural models was assessed by the eigenvalue index (i.e., the fraction

change in eigenvalues defined as Eq. (20)) and the RMSD index (i.e., the

root-mean-square-deviation of power spectral densities defined by Eq. (25)). Note that two

frequency bands 13.67 - 22.46 Hz and 40.04 - 49.81 Hz were utilized for the RMSD estimation.

Table 3 shows the estimated similarities of structural parameters (i.e., stiffness and damping) of the

three structural models (i.e., initial, stiffness-updated, and damping-updated models) as compared

to the experimental state of ‘the undamaged caisson with HWL’. It is observed that the eigenvalue

indices of the two modes converge as stiffness updated. It is also observed that the RMSD indices

273


converge as damping updated. Note that the convergence results verify the successful

identification of the structural model corresponding to ‘the undamaged caisson with HWL’.

5.2 Structural identification for water-level change The undamaged structure with LWL was analyzed for structural identification for water-level

change. It is noted that the water-level was varied from HWL to LWL to estimate its effect on

vibration characteristics as much as structural properties. The initial model was based on the

damping-updated model (see Table 2) for the undamaged caisson with HWL. By assuming that the

water level is known, the water height was adjusted as 0.296 m in the initial modeling. Therefore,

the total horizontal masses of considering the water level were estimated as: m1 = m3 = 21.25 kg

and m2 = 127.72 kg. As outlined in Table 4, the stiffness ratios of the initial model were the same

as the final damping-updated model for the undamaged caisson with HWL. Next, the first two

natural frequencies (ω1 = 19.78 Hz and ω2 = 44.68 Hz) and the corresponding damping ratios (ξ1 =

5.26 % and ξ2 = 2.64 %) were utilized for the calculation of the two Rayleigh damping coefficients

of Eq. (13).


Fig. 9 Vibration responses of damping-updated model for ‘the undamaged caisson with HWL’

Table 3 Convergence of structural models to ‘the undamaged reference caisson with HWL’

Eigenvalue index RMSD index

Mode 1 Mode 2 Mode 1 Mode 2

Initial 0.1955 0.5013 0.1373 0.0256

Stiffness update 0.0599 0.0235 0.1288 0.1330

Damping update 0.0599 0.0235 0.0247 0.0231

274


Table 4 Estimated structural parameters for ‘the undamaged caisson with LWL’


Initial

𝑘𝑆1 = 𝑘𝑆4 = 1.5𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.8𝑘𝐹2 [1.34 −0.89 0

−0.89 2.89 −0.890 −0.89 1.34

] × 106 [1.27 −0.59 0

−0.59 3.54 0.590 −0.59 1.27

] × 103

Stiffness

updated

𝑘𝑆1 = 𝑘𝑆4 = 5.76𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.43𝑘𝐹2 [1.49 −0.40 0

−0.40 2.13 −0.400 −0.40 1.49

] × 106 [0.30 −0.02 0

−0.02 1.49 −0.020 −0.02 0.30

] × 103

Damping

updated

𝑘𝑆1 = 𝑘𝑆4 = 5.76𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.43𝑘𝐹2 [1.49 −0.40 0

−0.40 2.13 −0.400 −0.40 1.49

] × 106 [1.19 −0.20 0

−0.20 3.81 −0.200 −0.20 1.19

] × 103

Table 4 outlines the calculated K and C matrices of the structural models in forms of relative

values. For the stiffness and damping updates, the procedures were performed the same as

described previously. Fig. 10 shows vibration responses of the initial model for the test case of

“the undamaged caisson with LWL”. Fig. 11 shows vibration responses of the stiffness-updated

model for the test case. Fig. 12 shows vibration responses of the damping-updated model for the

test case. From the figures, it is observed that the two acceleration signals the two PSD curves

become matched each other in the oscillating period, the peak frequencies and the dissipation

slopes.

Table 5 outlines the estimated similarities of structural parameters of the updated models as

compared to the experimental state of ‘the undamaged caisson with LWL’. It is observed that the

eigenvalue indices converge as stiffness updated and the RMSD indices converge as damping

updated. Note that the convergence results verify the successful identification of the structural

model corresponding to the water level change in the caisson system.


Fig. 10 Vibration responses of initial model for ‘the undamaged caisson with LWL’

275



Fig. 11 Vibration responses of stiffness-updated model for ‘the undamaged caisson with LWL’


Fig. 12 Vibration responses of damping-updated model for ‘the undamaged caisson with LWL’

Table 5 Convergence of structural models to ‘the undamaged caisson with LWL’



Initial 0.4545 0.0220 0.2863 0.2137


Damping update 0.0508 0.0220 0.1573 0.0521

276


As the water levels changed from HWL to LWL (see Fig. 3), i.e., the water height decreased by

13%, natural frequencies were increased and damping coefficients were slightly decreased as

outlined in Table 1. From the comparison of the identified structural model for the reference

structure (Table 2) to the corresponding one for the water-level variation into LWL (Table 4), it is

observed that system stiffness (indicated by K matrix) slightly increased and system damping

(indicated by C matrix) slightly decreased due to the water height decrement.

5.3 Structural identification for foundation damage The damaged structure with HWL was analyzed for structural identification for foundation

damage. It is noted that the foundation damage was inflicted into the test structure to estimate its

effect on vibration characteristics as much as structural properties. The initial model was based on

the damping-updated model (see Table 2) for the undamaged caisson with HWL. It was assumed

that the water level is known. The total horizontal masses, therefore, were computed for 0.34 m of

water level for the diagonal components as m1 = m3 = 22.95 kg and m2 = 138.06 kg ). As outlined

in Table 6, the stiffness ratios of the initial model were the same as the final damping-updated

model for the undamaged caisson with HWL. For the damping parameters, experimental modal

analysis of ‘the damaged caisson with HWL’ (ω1=15.39 Hz, ω2=42.24 Hz, ξ1=12.08%, ξ2=2.89%)

were applied as the initial values to calculate the damping parameters.

The model update was performed in the same way as described previously. Table 6 outlines the

calculated K and C matrices of the structural models in forms of relative values. Fig. 13 shows

vibration responses of the initial model for the test case of “the damaged caisson with HWL”. Fig.

14 shows vibration responses of the stiffness-updated model for the test case. Fig. 15 shows

vibration responses of the damping-updated model for the test case. From the figures, it is

observed that the two acceleration signals the two PSD curves become matched each other in the

oscillating period, the peak frequencies and the dissipation slopes.

Table 7 outlines the estimated similarities of structural parameters of the updated models as

compared to the experimental state of ‘the undamaged caisson with LWL’. It is observed that the

eigenvalue indices converge as stiffness updated and the RMSD indices converge as damping

updated. Note that the convergence results verify the successful identification of the structural

model corresponding to the foundation damage in the caisson system.


Fig. 13 Vibration responses of initial model for ‘the damaged caisson with HWL’

277



Fig. 14 Vibration responses of stiffness-updated model for ‘the damaged caisson with HWL’


Fig. 15 Vibration responses of damping-updated model for ‘the damaged caisson with HWL’

Table 6 Estimated structural parameters for ‘the damaged caisson with HWL’


Initial

𝑘𝑆1 = 𝑘𝑆4 = 1.5𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.8𝑘𝐹2 [

1.21 −0.80 0−0.80 2.60 −0.80

0 −0.80 1.21] × 106 [

0.42 0.15 00.15 3.4 0.15

0 0.15 0.42] × 103

Stiffness

updated

𝑘𝑆1 = 𝑘𝑆4 = 1.22𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.73𝑘𝐹2 [

1.31 −0.90 0−0.90 2.80 −0.90

0 −0.90 1.31] × 106 [

0.41 0.17 00.17 3.40 0.17

0 0.17 0.4] × 103

Damping

updated

𝑘𝑆1 = 𝑘𝑆4 = 1.22𝑘𝐹1

𝑘𝑆2 = 𝑘𝑆3 = 0.73𝑘𝐹2 [

1.31 −0.90 0−0.90 2.80 −0.90

0 −0.90 1.31] × 106 [

1.53 −0.68 0−0.68 5.14 −0.68

0 −0.68 1.53] × 103

278


Table 7 Convergence of structural models to ‘the damaged caisson with HWL’



Initial -0.1135 0.0235 0.0742 0.1110


Damping update 0.0 0.0235 0.0396 0.0494

As the foundation damage was simulated by 18.7% loss out of the bottom area (see Fig. 3),

natural frequencies were decreased and damping coefficients were increased as outlined in Table 1.

The updated structural model for the reference structure (Table 2) was compared to the

corresponding one for the foundation-damaged state (Table 6). It is observed that system stiffness

slightly decreased and system damping slightly increased due to the foundation damage.

6. Conclusions

A vibration-based structural identification method was proposed for the integrity assessment of

gravity-type caisson structures. A simplified structural model with a few degrees-of-freedom

(DOFs) was formulated to represent the gravity-type caisson structure that corresponds to the

sensors’ DOFs. Next, a structural identification algorithm based on the use of vibration

characteristics of the limited DOFs was formulated to fine-tune stiffness and damping parameters

of the structural model. At last, the experimental evaluation was performed on a lab-scaled

gravity-type caisson structure in a 2-D wave flume.

For structural identification of the caisson structure, structural models were sequentially

updated in three levels; i.e., initial, stiffness-updated, and damping-updated models. The

stiffness-updated model’s identification was successful in natural frequencies but different in

damping properties as compared to the experimental state. Also, the damping-updated model’s

identification was successful both in natural frequencies and damping properties.

As the water level was decreased by 13% into low water-level, two major results were analyzed

from vibration tests and structural identification as follows: first, natural frequencies were

increased and damping coefficients were slightly decreased due to the water height decrement; and

next, system stiffness slightly increased and system damping slightly decreased due to the water

height decrement. As the foundation damage was simulated by 18.7% loss out of the bottom area

of the foundation mound, two major results were analyzed as follows: first, natural frequencies

were decreased and damping coefficients were increased due to the occurrence of damage; and

next, system stiffness slightly decreased and system damping slightly increased due to the

foundation damage.

Acknowledgements

This work was supported by Basic Science Research Program through the National Research

279


Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology

(NFR-2013R1A1A2A10012040). The authors also would like to acknowledge the financial

support of the project ‘Development of inspection equipment technology for harbor facilities’

funded by Korea Ministry of Land, Transportation, and Maritime Affairs.

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